1. CHAPTER 2 Modeling Distributions of Data
2.1 Describing Location in a Distribution

2. Describing Location in a Distribution
FIND and INTERPRET the percentile of an individual value within a distribution of data.
ESTIMATE percentiles and individual values using a cumulative relative frequency graph.
FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.
DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.

3. Measuring Position: Percentiles
One way to describe the location of a value in a distribution is to tell what percent of observations are less than it.
The pth percentile of a distribution is the value with p percent of the observations less than it.
6 7
9 03
Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class?
Example
Her score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jenny is at the 84th percentile in the class’s test score distribution.
6 7
9 03

4. Cumulative Relative Frequency Graphs
A cumulative relative frequency graph displays the cumulative relative frequency of each class of a frequency distribution.
Age of First 44 Presidents When They Were Inaugurated
Age
Frequency
Relative frequency
Cumulative frequency
Cumulative relative frequency
40-44 2
2/44 = 4.5%
2/44 =
4.5%
45-49 7
7/44 = 15.9% 9
9/44 = 20.5%
50-54 13
13/44 = 29.5% 22
22/44 = 50.0%
55-59 12
12/44 = 34% 34
34/44 = 77.3%
60-64 41
41/44 = 93.2%
65-69 3
3/44 = 6.8% 44
44/44 = 100%

5. Since the mean is 80 and the standard deviation is about 6, Jenny’s score of 86 is about one standard deviation above the mean
Measuring z-Scores
Let’s return to the data from Mr. Pryor’s first statistics test, which are shown in the stemplot. 
The figure below provides numerical summaries from a computer output for these data. Where does Jenny’s score of 86 fall relative to the mean of this distribution? Because the mean score for the class is 80, we can see that Jenny’s score is “above average.”
But how much above average is it?
6 7
9 03

6. Measuring Position: z-Scores
A z-score tells us how many standard deviations from the mean an observation falls, and in what direction.
If x is an observation from a distribution that has known mean and standard deviation, the standardized score of x is:
A standardized score is often called a z-score.
Example
Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is What is her standardized score?

7. PROBLEM: Find the (z-scores) for each of the following students in Mr
PROBLEM: Find the (z-scores) for each of the following students in Mr. Pryor’s class. Interpret each value in context.
(a) Katie, who scored 93.
(b) Norman, who earned a 72.
SOLUTION:
(a) Katie’s 93 was the highest score in the class. Her corresponding z-score is
In other words, Katie’s result is 2.14 standard deviations above the mean score for this test.
(b) For Norman’s 72, his standardized score is
Norman’s score is 1.32 standard deviations below the class mean of 80.

8. Transforming Data
Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution.
Effect of Adding (or Subtracting) a Constant
Adding the same number a to (subtracting a from) each observation:
adds a to (subtracts a from) measures of center and location (mean, median, quartiles, percentiles), but
Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation).

9. Transforming Data
Example
Examine the distribution of students’ guessing errors by defining a new variable as follows:
error = guess − 13
That is, we’ll subtract 13 from each observation in the data set. Try to predict what the shape, center, and spread of this new distribution will be. n
Mean sx
Min Q1 M Q3
Max
IQR
Range
Guess(m) 44
16.02
7.14 8 11 15 17 40 6 32
Error (m)
3.02 -5 -2 2 4 27

10. Transforming Data
Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution.
Effect of Multiplying (or Dividing) by a Constant
Multiplying (or dividing) each observation by the same number b:
multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b
multiplies (divides) measures of spread (range, IQR, standard deviation) by |b|, but
does not change the shape of the distribution

11. Transforming Data
Example
Because our group of Australian students is having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 to 3 meters too high. Let’s convert the error data to feet before we report back to them. There are roughly 3.28 feet in a meter. n
Mean sx
Min Q1 M Q3
Max
IQR
Range
Error (m) 44
3.02
7.14 -5 -2 2 4 27 6 32
Error(ft)
9.91
23.43
-16.4
-6.56
6.56
13.12
88.56
19.68
104.96

12. Describing Location in a Distribution
FIND and INTERPRET the percentile of an individual value within a distribution of data.
ESTIMATE percentiles and individual values using a cumulative relative frequency graph.
FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.
DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.